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Investigation of Cascadia Tremor Correlation with Earth Tides
Carl W. Ulberg
Senior Integrative Exercise March 9, 2007
Submitted in partial fulfillment of the requirements for a Bachelor of Arts degree from Carleton College, Northfield, Minnesota
Table of Contents Abstract Introduction 1 Geologic Setting 2 Cascadia Subduction Zone 3 Previous Work on Tremor in Cascadia 7 Methods 8 Studying Tremor 8 Instrument Deployment 8 Examination of Seismic Data 9 Examination of Possible Tremor Causes 11 Ocean Tides Data Set 11 Earth Tides Data Set 11 Comparison of Earth Tides and Ocean Tides 15 Comparison of Earth Tides and Ocean Tides with Tremor 21
Results 22 Tremor 22 Comparison of Tremor with Tides 23 Discussion 31 Previous Work on Correlating Earthquakes with Earth Tides 31 Discussion of the Results of this Study 33 Possibilities for Future Study 34 Conclusions 35 Acknowledgements 36 References Cited 37 Appendix A: Stress Tensor Calculations 40 Appendix B: Tremor events for 2006 41 Appendix C: Matlab Code 42
Investigation of Cascadia Tremor Correlation with Earth Tides
Carl W. Ulberg Carleton College
Senior Integrative Exercise March 9, 2007
Advisors:
Kenneth C. Creager, University of Washington Sarah J. Titus, Carleton College
ABSTRACT
Major episodes of episodic tremor and slip located along the Cascadia subduction zone occur every ~14 months and have been well-studied using both seismic and geodetic data. I look at the interval between major events using vertical component data from 11 Pacific Northwest Seismograph Network seismometers broadly distributed across western Washington and find that there are still tremor episodes, although of smaller duration and magnitude. Typical smaller tremor bursts occur over 2 to 3 days, containing periods of up to 8 continuous hours exhibiting similar waveforms, and appear in 2 to 8 stations at a time. I compare the occurrence of tremor episodes with Coulomb stresses due to Earth tides and ocean tidal loading, and find a possible correlation between the two data sets. Keywords: Earth tides, Cascadia subduction zone, earthquakes, seismology, correlations
INTRODUCTION
Cascadia subduction zone non-volcanic tremor is a recently discovered
phenomenon (Dragert et al., 2001). It can be identified by unique non-earthquake
seismic signals associated with slow slip events along the deeper portion of the
subduction interface (Dragert et al., 2001; Rogers and Dragert, 2003). Tremor has also
been observed in the Nankai subduction zone in Japan and along the San Andreas fault of
California (Obara, 2002; Johnston, 2006). The exact mechanism behind tremor is
unknown, but one possibility is the movement of fluid in the subduction zone due to the
dehydration of the subducting slab (Obara, 2002). Tremor has been associated with
changes in plate motion as measured using GPS data. It is important to study tremor
because it is believed that it can add to stress in the locked portion of a plate boundary
interface, thereby increasing the chance of a large, damaging earthquake (Rogers and
Dragert, 2003).
The first evidence for tremor in the Cascadia subduction zone was found in
Washington State and Vancouver Island, Canada, in 2001 (Dragert et al., 2001). Since
then, it has been determined by Dragert et al. (2001) and others that major tremor bursts
occur along the Cascadia subduction zone every ~14 months. Each burst lasts for
approximately two weeks, during which the tremor occurs almost continuously over a
broad 50-100 km area. GPS data compared with these tremor bursts show that there is
slip in the movement of the North American plate associated with tremor (Rogers and
Dragert, 2003; Dragert et al., 2001). This process is known as episodic tremor and slip.
1
In this study I examine tremor in the state of Washington for the year of 2006
using seismic data. During this year there were no large episodic tremor and slip events.
However, there were smaller bursts of tremor, and I compare the occurrence of these
tremor events with tidal cycles, for both ocean tides and Earth tides. Specifically, I seek
to determine whether the stresses on the Earth due to ocean and earth tides are large
enough to trigger a tremor episode, and if tremor occurs at times when the stresses are
aligned correctly. This can help to better determine the mechanisms for causing tremor.
GEOLOGIC SETTING
Cascadia Subduction Zone
The Cascadia subduction zone results from the oblique subduction of the oceanic
Juan de Fuca plate beneath the continental North American plate (Fig. 1). This
subduction zone has been active since the Mesozoic, beginning with the Farallon plate
subducting beneath North America (Wilson, 1998). At approximately 30 Ma, the Juan de
Fuca plate separated from the rest of the Farallon plate along the Pioneer fracture zone,
which is near the Mendocino fracture zone shown in Figure 1 (Wilson, 1988). Since 30
Ma, the convergent margin has been rotating clockwise due to being caught in the larger
dextral shear boundary between the North American and Pacific plates (Wells et al.,
2002).
For the last 20 Ma, convergence between the Juan de Fuca and North American
plates has been relatively constant at a rate of ~40 mm/yr in a N55°W direction (Wilson,
2002). Currently, convergence rates change along strike from ~30 mm/yr at 42°N to ~45
2
Figure 1. Tectonic context of the Cascadia subduction zone, from Dragert et al. (1994). The Cascadia subduction zone occurs where the oceanic Juan de Fuca plate subducts beneath the continental North American plate. Square box indicates Figure 2 map location and bold line indicates Figure 3 cross-section location.
Fig. 2
Fig.3
3
mm/yr at 49°N, although these along-strike differences have changed through time
(DeMets et al., 1990).
Subduction of the Juan de Fuca plate is relatively shallow compared to many
subduction zones, with a dip angle of ~10° in the first 50 km of depth (Crosson and
Owens 1987). There is an arch in the plate near the latitude of the Puget Sound, causing
a bend of ~45° in the strike of the plate (see plate depth contours in Fig. 2). South of this
bend the Juan de Fuca plate strikes almost due north, and north of this bend the plate
strikes in a N45°W direction. This bend is attributed to a geometric space problem
caused by the Juan de Fuca plate subducting into a concave oceanward corner of the
North American plate. (Crosson and Owens, 1987; Trehu et al., 2002; Chiao and Creager,
2002)
The Cascadia subduction zone has ruptured periodically (every ~600 yrs) in giant
megathrust earthquakes (Mw>8) throughout the Holocene (Atwater, 1987; Satake et al.,
1996). These earthquakes are interpreted to occur on the locked portion of the interface
between the Juan de Fuca and North American plates (Fig. 3). This zone is located at
depths of ~10 km and temperatures of ~350°C, from 40 to 90 km off the coast of the
western United States (Hyndman, 1997). The last major thrust earthquake occurred in
1700, and is believed to have ruptured the entire Cascadia convergent margin, resulting in
a tsunami which reached as far away as Japan (Atwater, 1987; Satake et al., 1996).
Intraslab earthquakes occur within the subducting Juan de Fuca plate, and have a
larger potential for causing damage in North America than offshore megathrust
earthquakes (Preston et al., 2003). This is because they can occur directly underneath the
continent and beneath major urban centers. For example a Mw=6.8 earthquake ruptured
4
-124° -122° -120°
46°
47°
48°
49°
100 km
CAFE BBCAFE arraysPNSN BBVolcano
20 k
m
40 k
m
60 k
m
80 k
m
Figure 2. Map showing the CAFE deployment as well as the PNSN broadband stations used for this study. Depth contours (thick yellow lines) of the subducting Juan de Fuca plate are shown every 20 km. Modified from Abers et al. (2007).
La Push
Victoria
Sequim
Hoodsport
5
z
Trench Victoria
N. AMERICA PLATE
Cross-section viewed from south
TransientETS Release
LongtermLocked-Zone
Push
Figure 3. Cross-section of the Cascadia subduction zone near Vancouver Island, showing basic mechanisms of episodic tremor and slip. Approximate tremor locations are indicated by stars. From Geological Survey of Canada.
6
underneath Nisqually in the south Puget Sound in 2001, causing much property damage
to the greater Puget Sound area, including Seattle (Preston et al., 2003).
Previous Work on Tremor in Cascadia
Tremor in Cascadia occurs primarily between 25 and 35 km in depth, although it
can range from 11-48 km (Rogers and Dragert, 2003; Kao et al., 2005). Much of the
tremor occurs near the interface between the Juan de Fuca plate and the North American
plate, both within the subducted slab and in the overriding crust (Fig. 3; Kao et al., 2005).
Locations of tremor are primarily under the Puget Sound and the southeast tip of
Vancouver Island, above the 40 km depth contour of the Juan de Fuca plate (Rogers and
Dragert, 2003; Kao et al., 2005), although they can also occur as far south as northern
California (McCausland et al., 2005). Figure 3 shows the basic mechanisms of episodic
tremor and slip in Cascadia.
The frequency content of tremor as a seismic signal is in the range between 1 and
6 Hz. This is lower than the frequency range found for small earthquakes, which is
normally above 10 Hz (Rogers and Dragert, 2003; Kao et al., 2005). A tremor signal is
usually emergent rather than impulsive (meaning that it increases gradually without any
clear starting point), and can last anywhere from several minutes to several hours.
Tremor signals are strongest on horizontal seismographs and are thought to move at shear
wave velocities of ~4 km/s (Obara, 2002; Rogers and Dragert, 2003). In order to be
classified as tremor, a tremor-like signal must be seen on multiple seismographs in a
network, with each signal exhibiting a similar shape (Rogers and Dragert, 2003).
7
Otherwise, noise from wind or cultural sources at a particular seismogram could be
mistaken for tremor.
The exact cause of tremor is unknown, but one possibility is the movement of
fluid in the subduction zone, possibly related to the dehydration of the subducting slab
(Obara, 2002; Kao et al., 2005). Dragert et al. (2001) and others have theorized that
tremor episodes can add to the cumulative stress on the locked portion of the Juan de
Fuca-North American plate interface, increasing the probability of a large, megathrust
earthquake.
METHODS
Studying Tremor
Instrument Deployment
This project began as part of an IRIS internship at the University of Washington,
supervised by Ken Creager. My involvement for the internship included helping to scout
locations and deploy 62 broadband seismometers around the western part of the state of
Washington during the summer of 2006 (Fig. 2). This deployment was for the CAFE
(Cascadia Array For Earthscope) experiment, which is part of the Earthscope initiative.
Some of the requirements for selecting a site for a seismometer were: open sky
(for solar panels), some distance (>1 km) from major roads and cities (to reduce noise
and to lessen the risk of vandalism), and reasonable proximity to smaller roads (for
transport of seismometer, batteries, cement, etc.). For these reasons, many of the
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seismometers were placed in clearcuts owned by timber companies, although there were
some on private or government property.
Examination of Seismic Data
For this study, I examined data from eleven seismometers broadly distributed
across western Washington and northwest Oregon (Fig. 2). These seismometers are
similar to the ones that were deployed for the CAFE experiment, but are maintained by
the Pacific Northwest Seismograph Network. Data were collected for the year of 2006,
although due to winter conditions, four different stations were used starting in November.
I looked at the vertical component data from broadband or short-period instruments that
were band-pass filtered at 1.5 to 5 Hz (The data were run through a Fourier transform,
and the high- and low-frequency signals were discarded; the data were then run back
through an inverse Fourier transform to enhance the seismic signal over periods of 0.2-
0.7 s). An envelope function (envelope functions emphasize times when seismic energy
is highest by essentially squaring the signal) calculated from the resulting waveforms was
then low-pass filtered at 0.05 Hz (this enhances the signal over periods of 1/.05=20
seconds).
A tremor episode was classified as an emergent increase in seismic activity across
multiple stations lasting from 10 minutes to several hours. Tremor must be observed
with similar waveforms at different stations since a signal at a single station could likely
be the result of noise from wind or cultural sources. For this study, observations were
done by hand. Figure 4 shows an example of data from the eleven seismometers where
9
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60.
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or in
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rem
or.
10
tremor was identified at 4 stations (stations 3, 6, 7, and 8). Note the similarity in the
shapes of the waveforms.
Once a tremor episode was identified, the starting time (all times used in this
study are reported as Greenwich Mean Time), duration, and stations in which it was
present were recorded. At the end of the year, this data was combined into one time
series containing the number of hours of tremor that occurred during each day of the year.
Consecutive days exhibiting tremor were classified as single ‘events.’
Examination of Possible Tremor Causes
Ocean Tides Data Set
Predicted ocean tide heights were downloaded for the tidal gauges at La Push,
WA and Sequim, WA (Fig. 2), and from the NOAA (National Oceanic and Atmospheric
Administration) website. These two locations give the tides along the Pacific coast of
Washington as well as within the Strait of Juan de Fuca. Predicted ocean tides were also
obtained using software from Agnew (1996), assuming the tidal model of Schwiderski
(1980). This method yields similar results to the NOAA data.
In order to determine how the ocean tides changed over the course of the year, the
tidal range (the difference between maximum and minimum tide) was computed for each
day of 2006.
Earth Tides Data Set
Data to study Earth tides were obtained from software by Agnew (1996). This
software calculates stresses at the surface of the Earth due to Earth tides as well as ocean
11
tidal loading, in the form of a stress tensor. Stress tensors are useful because they can be
used to obtain a traction vector on a plane of interest (in this case, the plane
approximating the Juan de Fuca plate under Washington state). Traction is defined as
force per unit area (the same as stress), and can be decomposed into three components:
(1) a component in the direction normal to the plane, (2) a component within the plane in
question and in the direction of slip on the fault, and (3) a component within the plane in
question and perpendicular to the previous two. Table 1 shows the vectors used to
describe the Juan de Fuca plate. The two components of the traction vector that are used
for this study are in the normal direction and in the slip direction. The magnitudes of
these vectors are equal to the normal and shear stresses on the plane, respectively. See
Appendix A for a detailed explanation of these calculations.
Table 1. Vectors describing the Juan de Fuca plate. Vector Components Strike of plate (v1) <cos a, sin a, 0> Dip of plate (v2) <-sin a cos b, cos a cos b, sin b> Normal to plate (v3) <sin a sin b, -cos a sin b, cos b> In direction of plate
motion (vslip) slip=<cos(dir) sin(dir) 0> -direction of plate motion at surface vslip=slip-dot(slip,v3)*v3; -projection onto plane
With a as the strike angle and b as the dip angle of the subducting plate, and dir as the direction of slip (Fig. 5). In calculating vslip, dot() signifies dot product. The coordinate system used is x=North, y=East, z=down Stress tensors were obtained at 30-minute intervals at four points above the Juan
de Fuca plate (Table 2). These sites were chosen because of their positions above the
subducting plate with different strikes, as determined from a depth contour plot similar to
Figure 2. Three points are on land near the projection of the 40 km slab depth contour,
the primary zone of tremor occurrence, and one point (La Push) is off the coast of
Washington to provide a reference point for ocean tidal loading.
12
v1 (strike)
v2 (dip)
v3 (normal)
Down
North
East
South
West
Up
vslip
Figure 5. Geometry of a plane representing the Juan de Fuca plate. The plane can be described by three vectors: v1 (in the direction of strike), v2 (in the dip direction), and v3 (normal to the plane). The vector in the direction of slip (vslip) is within the plane. A point on the Juan de Fuca plate where it strikes approximately N15E as in this figure would be found in southern Washington.
a
b
13
Table 2. Sites for tensor calculations (see Fig. 2 for map locations) Site Latitude
(°N) Longitude (°W)
Approximate Strike of Juan de Fuca Plate (°)
Approximate Depth of Juan de Fuca plate (km)
La Push, WA 47.90 124.65 315 15 Victoria, BC 48.44 123.39 315 40 Sequim, WA 48.04 123.13 345 40 Hoodsport, WA 47.51 123.08 0 40 In order to quantify the change in shear and normal stresses at these points, a
Coulomb stress analysis was used, as described by Cocco and Rice (2002). Shear and
normal stresses can be used to determine Coulomb stress changes on a given fault using
the following relation: failure due to frictional faulting occurs when the shear stress
exceeds the coefficient of friction multiplied by the effective normal stress, or
σs ≥ μ(-σn – p), (1)
where σs is shear stress (in the direction of slip), σn is normal stress (positive for
extension), μ is the coefficient of friction of the rocks (typically between 0.6 and 0.8;
Harris, 1998), and p is pore pressure. Although pore pressure has a large effect on
Coulomb stress calculations, it is ignored in this study by assuming that it does not
change. Additionally, a value for μ of 0.6 was used for this study.
This relation (1) can be better quantified by the Coulomb stress, σs + μσn. Large
values for the Coulomb stress correspond to the largest probability of fracture along the
Cascadia subduction interface. It is important to note that the stresses on the Juan de
Fuca fault are modulated by many factors besides Earth tides, but this study focuses on
the change in Coulomb stress due to tides. To calculate a stress time series to compare to
tremor events, the maximum tidal Coulomb stress was calculated for each day of 2006.
14
Comparison of Earth Tides and Ocean Tides
Before comparing ocean and Earth tides with tremor, it is important to understand
how these two data sets relate to each other. To do this, I plotted the ocean and Earth
tides together on daily and monthly scales. Figures 6 and 7 show the ocean tide height
plotted for one day with shear, normal, and Coulomb stress on the Juan de Fuca plate at
two points used for this study, La Push and Sequim. The stresses at La Push were
calculated several miles off the coast at the bottom of the ocean, whereas the stresses at
Sequim were calculated several miles inland at the surface. This was done to examine
the effect of ocean tidal loading on the data sets.
The plots appear somewhat similar due to diurnal or semi-diurnal tides, with two
peaks and two troughs a day, with one peak possibly more pronounced. However, there
are several obvious differences, which can be attributed to the effect of ocean tidal
loading. First, the amplitude of the shear and normal stresses at La Push are orders of
magnitude larger than the stresses at Sequim, due to the fluctuations in tidal height (a 1 m
displacement of water equates to 1.00*103 kg/m3*9.8 m/s2*1 m ≈ 10 kPa). Second, peaks
(or troughs) in stresses at Sequim are shifted by 1-2 hours in relation to the ocean tide,
whereas the stresses and ocean tide at La Push line up almost exactly. This is again due
to the large effect of ocean tidal loading at La Push. Third, the sign of the normal stress
switches between Sequim and La Push- at La Push, the shear and normal stresses have
the same sign, whereas at Sequim the shear and normal stresses have opposite signs. The
sign of the normal stress at La Push can be explained by high ocean tide corresponding to
the maximum compression on the Earth from ocean tidal loading. Because extensional
stress is positive in this stress data set, this would explain the negative (compressive)
15
305 305.5 306
0
0.5
1
Tide
hei
ght (
m)
305 305.5 306
0
0.5
1x 10 4
Stre
ss (P
a)
Day of 2006
Figure 6. Comparison of ocean tide with stresses due to Earth tides and ocean tidal loading at La Push, WA for November 1, 2006. (A) Ocean tidal height. (B) Calculated shear (solid red), normal (dashed green), and Coulomb stress (dot-dashed black) on the Juan de Fuca plate.
A
B
-0.5
-1
-0.5
-1
16
305 305.5 306
0
0.5
1
Tide
hei
ght (
m)
305 305.5 306
0
200
Stre
ss (P
a)
Day of 2006
Figure 7. Comparison of ocean tide with stresses due to Earth tides and ocean tidal loading at Sequim, WA for November 1, 2006. (A) Ocean tidal height. (B) Calculated shear (solid red), normal (dashed green), and Coulomb stress (dot-dashed black) on the Juan de Fuca plate.
A
B
-0.5
-200
-1
17
normal stresses during high tide. If the Sequim tensor were calculated at a point with
fluctuating levels of water above it, it would have a similar behavior, and conversely, if
the tensor calculated at La Push were at a point on land, it would resemble the results
from Sequim. Indeed, when a tensor was calculated at the surface at La Push, it exhibited
similar characteristics to the Sequim stresses.
Figures 8 and 9 show the ocean tide height plotted with shear, normal, and
Coulomb stress for the month of November, at La Push and Sequim. Shear and normal
stresses vary on a spring/neap cycle at the same times as ocean tides. Spring tide, when
these values have the largest range, occurs when the moon and Sun are in alignment with
the Earth (during new and full moons, also known as syzygy). Additionally, neap tide,
when these values have the smallest range, occurs when the moon and Sun are out of
alignment with the Earth (during the first and third quarter of the moon, also known as
quadrature).
Based on examination of Figures 6-9, maximum Coulomb stresses occur near low
ocean tides, when the sum of the shear and normal (extensional) stresses on the Juan de
Fuca plate interface is at a maximum. These stresses are greater under the ocean, but still
exhibit the same qualitative characteristics on land, due to the low value for normal
stress. Furthermore, maximum Coulomb stresses will be greatest during spring tides,
when there are the lowest low tides. These results are supported by Wilcock (2001), who
found that earthquakes occur more frequently along the Juan de Fuca Ridge near low
tides, and especially the lowest spring tides, attributed to maximum extensional stresses
in all directions.
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305 310 315 320 325 330 335
0
1
2
Tide
hei
ght (
m)
305 310 315 320 325 330 335
0
1
2x 10 4
Stre
ss (P
a)
Day of 2006
Figure 8. Comparison of ocean tide with stresses due to Earth tides and ocean tidal loading at La Push, WA for the month of November, 2006. (A) Ocean tidal height. (B) Calculated shear (solid red), normal (dashed green), and Coulomb stress (dot-dashed black) on the Juan de Fuca plate.
A
B
19
305 310 315 320 325 330 335
0
1
2
Tide
hei
ght (
m)
305 310 315 320 325 330 335
0
500
Stre
ss (P
a)
Day of 2006
-1
-2
-500
A
B
Figure 9. Comparison of ocean tide with stresses due to Earth tides and ocean tidal loading at Sequim, WA for the month of November, 2006. (A) Ocean tidal height. (B) Calculated shear (solid red), normal (dashed green), and Coulomb stress (dot-dashed black) on the Juan de Fuca plate.
20
Comparison of Earth Tides and Ocean Tides with Tremor
In order to compare tremor with earth and ocean tides, I used the following
hypothesis: tremor occurs as slip along the interface between the Juan de Fuca and North
American plates, and this slip is most likely to occur when the Coulomb stress due to
Earth tides in that orientation is at its greatest. On a daily scale, the Coulomb stress due
to Earth tides is greatest during low ocean tide, so this hypothesis would predict more
tremor occurring during low ocean tides on a daily scale. On a monthly scale, Coulomb
stresses due to Earth tides are greatest during spring tide (that is, the lowest low tides and
their associated large Coulomb stresses occur during spring tide), so the hypothesis
predicts more tremor occurring during spring tide on a monthly scale. For this study, I
focused on the monthly scale, using two methods to determine a correlation between
spring tide and tremor occurrences. To perform correlation tests between tides and
tremor, I used Matlab version 7.3.0.267 (R2006b). See Appendix C for the code.
For the first test, cross-correlations were performed between tremor and ocean
tidal range (as calculated at La Push and Sequim, WA), as well as between tremor and
maximum Coulomb stress (as calculated at La Push, Sequim, Victoria, BC, and
Hoodsport). This method gives a measure of the similarity between two data sets. The
output is a third time series with length 2N + 1, where N is the length of the two original
series (in this case, N=365 days). Each point on the new series represents the correlation
at a certain lag between the two original series when they have been shifted that many
days with respect to each other. A time series cross-correlated with itself would give a
maximum correlation value at a lag of 0 days. In this analysis, a peak correlation at 0 lag
21
would indicate that tremor is well-correlated with either maximum ocean tidal range
and/or maximum Coulomb stress.
The second test for correlation between tremor and tides is similar to that
described by McNutt and Heaton (1981). This test divides the year into bins representing
days before or after peaks (local maxima) in the tidal time series (in either ocean tidal
range or maximum Coulomb stress). Each day of the year was assigned to a bin
depending on how far removed it was from the closest tidal maximum. Then the number
of hours of tremor and days with tremor were determined for each bin. Since maxima in
ocean tidal range or Coulomb stresses did not occur at regular intervals, the bins near the
edge (at +/- 8 days) have fewer possible days or hours with tremor.
RESULTS
Tremor
Tremor during the year 2006 can be characterized as a series of small tremor
events (relative to the large 14-month episodic tremor and slip events), each usually
lasting two to three days and typically containing 5-10 hours of tremor a day, with
extremes of 0.5-18 hours in one day. A total of 259.5 hours of tremor occurred during 77
days of 2006, and these were divided into 38 tremor events, ranging from 0.5 hours in
one day to 61 hours over five days (see Appendix B for a listing of tremor events). The
number of stations that recorded tremor ranged from two to eight for each tremor
episode. There was usually a period of one to two weeks between these tremor episodes,
22
although a period of 20 days without tremor was recorded in October. Figure 10 shows
tremor events for 2006.
Comparison of Tremor with the Tides
A preliminary comparison of tremor with ocean tidal range and maximum
Coulomb stress is shown in Figure 11, in which the time series are simply plotted
together. There is no obvious correlation between the data sets, although it appears that
many tremor bursts occurred at or near times of peak tidal range and maximum Coulomb
stresses. Of course, there were also tremor episodes that occurred during low tidal ranges
or low variations in maximum Coulomb stresses.
Cross-correlations of tremor with ocean tidal range (Fig. 12) and maximum
Coulomb stress (Fig. 13) generally show a peak near 0 lag. This indicates that the data
sets are best correlated when shifted by <2 days, which means that peaks of tremor
generally occur at the same times as peaks in ocean tidal range and maximum Coulomb
stresses. The peaks are repeated at intervals of approximately 14 and 28 days because of
the inherent cyclicity of the tides. This repetition of peaks means that in order to read
these figures, one must focus on the interval between shifts of –14 and +14 days. In
Figure 13A, there is a clearer peak near 0 lag for the cross-correlation between maximum
Coulomb stress and tremor at the La Push site as compared to the three other sites. In
fact, the cross-correlation at Victoria has a dominant peak at –6 days away from 0 lag
(Fig. 13B), and the cross-correlation at Sequim has a peak at +1 lag that is lower than the
peak at –9 lag (Fig. 13C).
23
50 100 150 200 250 300 3500
2
4
6
8
10
12
14
16
18
20
Day of 2006
Trem
or h
ours
Figure 10. Hours of tremor per day for the year 2006.
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50 100 150 200 250 300 3500
2
4
6
8
10
12
14
16
18
20
Day of 2006
Trem
or h
ours
Figure 11. Tremor (black bars) plotted with ocean tidal range as measured at Sequim (solid blue line) and maximum daily Coulomb stress as measured at Hoodsport (dashed red line) for the year 2006.
0
60
120
180
240
300
1
2
3
4
Maxim
um C
oulomb
stress (Pa)
Ocean Tidal R
ange
25
0 20 40 60
0
0.1
0.2
0 20 40 60
0
0.1
0.2
Time (days)
Figure 12. Cross-correlation of tremor with ocean tidal range at La Push (A) and Sequim (B), WA. The x-axis represents the number of days the time series were shifted to get the given correlation. The y-axis scale is meaningless, other than to say that the series are better correlated at higher values.
-0.1
-0.1
-0.2
-0.2
-60
-60
-40
-40
-20
-20
B
A La Push
Sequim
26
0 50
0
0.1
0.2La Push
0 50
0
0.1
0.2Victoria
0 50
0
0.1
0.2Sequim
Time (days)0 50
0
0.1
0.2Hoodsport
Figure 13. Cross-correlation of tremor with maximum daily Coulomb stress at La Push (A), Victoria (B), Sequim (C), and Hoodsport (D). The x-axis represents the number of days the time series were shifted to get the given correlation. The y-axis scale is meaningless, other than to say that the series are better correlated at higher values.
-0.1 -0.1
-0.1-0.1
-0.2
-0.2
-0.2
-0.2
-50
-50 -50
-50Time (days)
A B
C D
27
Figures 14A and 14B show the results of plotting the number of tremor hours and
days with tremor as a function of ocean tidal range at La Push and Sequim, respectively.
Both exhibit peaks in days and hours with tremor when the tidal range was near its
maximum, above the amount that would be expected if tremor were distributed
uniformly. As shown in Table 3, the probabilities of getting these peaks of tremor days
near a maximum tidal range with a uniform distribution of tremor are 0.21 and 0.09 for
the series at La Push and Sequim, respectively.
Figure 15 shows the number of tremor hours and days in relation to maximum
Coulomb stress at the four sites used for this study. Similar to Figure 14, Figure 15A
shows a peak near 0 with a probability of only 0.09 of being obtained with a uniform
distribution of tremor. Figures 15B-D, on the other hand, do not show pronounced peaks
of tremor days or hours near days with the highest Coulomb stresses, and have
probabilities of 0.81, 0.40, and 0.51 of occurring with uniformly distributed tremor.
DISCUSSION
Previous Work on Correlating Earthquakes With Earth Tides
There are many previous studies that investigated correlations between
earthquakes and earth tides (Emter, 1997), with positive (Wilcock, 2001), negative
(McNutt and Heaton, 1981; Vidale, 1998), and mixed results (Tsuruoka et al., 1995).
This study seeks to perform the same types of comparisons using tremor and Earth tides,
by assuming that tremor is equivalent to slip on a fault with the same depth and geometry
as the subducting Juan de Fuca plate. This is a reasonable assumption, since previous
28
-8 -6 -4 -2 0 2 4 6 8
La Push
-8 -6 -4 -2 0 2 4 6 8
Sequim
Days from peak tidal range
0
10
20
30
40
50H
ours of tremor
0
10
20
30
40
50
Hours of trem
or
0
3
6
9
12
15
Day
s w
ith tr
emor
0
3
6
9
12
15
Day
s w
ith tr
emor
Figure 14. Histograms showing the distribution of days (blue) and hours (red) of tremor with respect to the day of a peak ocean tidal range, as recorded at La Push (A) and Sequim (B). The dotted line indicates the number of hours or days of tremor expected in a uniform tremor distribution. This line is not horizontal because the peaks of tidal ranges did not occur at regular intervals- days between peaks ranged between 9 to 17 days. Therefore, the bins on the edges did not occur as frequently.
B
A
29
Figure 15. Histograms showing the distribution of days (blue, left bar) and hours (red, right bar) of tremor with respect to the day of a peak Coulomb stress, as recorded at La Push (A) Victoria (B), Sequim (C), and Hoodsport (D). The dotted line indicates the number of hours or days of tremor expected in a uniform tremor distribution. This line is not horizontal because the peaks of Coulomb stress did not occur at regular intervals- days between peaks ranged between 9 to 17 days. Therefore, the bins on the edges did not occur as frequently.
-5 0 5
La Push
-5 0 5
Victoria
-5 0 5
Sequim
Days from peak Coulomb stress-5 0 5
Hoodsport
Days from peak Coulomb stress
0
10
20
30
40
50H
ours of Tremor
0
10
20
30
40
50
Hours of Trem
or
0
3
6
9
12
15
Day
s of
Tre
mor
0
3
6
9
12
15
Day
s of
Tre
mor
A B
C D
30
Table 3. Probabilities that a uniform distribution of tremor would match actual data. Figure number 14A 14B 15A 15B 15C 15D Expected days of tremor in –1, 0, +1 bins
16.7 16.7 16.7 16.7 16.7 16.7
Probability p of a random day being in one of these bins
0.22 0.22 0.22 0.22 0.22 0.22
Standard deviation √[np(1-p)], n=77 3.61 3.61 3.61 3.61 3.61 3.61 Actual days of tremor in –1, 0, +1 bins
20 22 22 14 18 17
P(X ≥ actual) = (2) 0.21 0.09 0.09 0.81 0.40 0.51 Note: To test the significance of the peaks for tremor days, I assumed that tremor occurred randomly in 2006 and determined how likely it would be to see these distributions if that were the case. Specifically, I test the null hypothesis that tremor occurs randomly by assuming that tremor behaves like a binomially distributed random variable X. A binomial distribution is the probability distribution of the number of successes in a sequence of n Bernoulli trials, each with a probability p of succeeding. Bernoulli trials are simply trials that can either succeed or fail (in this case, tremor can occur or not occur). Since tremor occurred during 77 days in 2006, n = 77. The probability p of a random tremor event occurring during one of the three central bins varies between 0.22 and 0.28. In a binomial distribution, the probability that the random variable X equals a value k is equal to P(X = k) = nCk*pk(1- p)n-k [nCk signifies ‘n choose k,’ which is equivalent to n!/(k!(n-k)!)]. Therefore, the probability that tremor occurs at least a certain number of times x is P(X ≥ x) = 1 – P(X < x) =
1- (2) knkx
kpp
kn −
−
=
−⎟⎟⎠
⎞⎜⎜⎝
⎛∑ )1(1
0
31
work has shown that many tremor events occur at depths near the subduction interface
(Rogers and Dragert, 2003; Kao et al., 2005). Studies in Japan have also shown that low-
frequency earthquakes, which are closely associated with tremor, have focal mechanism
solutions with nodal planes consistent with the dip of the subducting Philippine Sea plate
(Shelly et al., 2006; Ito et al., 2007).
Earth tides, much like the ocean tides, are the result of the gravitational
interaction between the Earth, Moon, and Sun, which deforms the solid Earth (Melchior,
1983). Average solid Earth tide stresses in the continents have amplitudes of 1-4 kPa,
whereas Earth tide stresses in the oceans can reach 10 kPa, due to ocean tidal loading
(Figures 6-9; Melchior, 1983; Wilcock, 2001). These values are above the average rate
of tectonic stress buildup (10-100 Pa/hr or much lower- in Cascadia, where megathrust
earthquakes have recurrence intervals of ~500 years [~5*106 hours], the average tectonic
stress can be calculated by the stress drop of a typical megathrust earthquake divided by
the recurrence interval, 1-10 MPa/5*106 hours ≈ .2-2 Pa/hr), but are well below the
average stress drop in earthquakes (1-10 MPa; Kanamori and Anderson, 1975). They are
also well below the average lithostatic pressure in the Earth at the depths that this study
focuses on. For example, the pressure on the Earth at a depth of 40 km is ~1.1 GPa,
assuming a crustal density of 2.83*103 kg/m3.
Although stresses due to Earth tides are orders of magnitude smaller than stress
drops from earthquakes, the fact that Earth tide stresses are greater than average rates of
tectonic stress buildup suggests that earth tides can trigger tremor episodes. If average
tectonic stress buildup in the Cascadia subduction zone is around 1 Pa/hour, then a
modulation of this value by Earth tide stresses with amplitudes of ~1-10 kPa could cause
32
a critical stress amount to be reached before the average tectonic buildup reached that
point (Vidale et al., 1998; Thatcher, 1982).
Discussion of the Results of this Study
The cross-correlations performed between tremor and ocean and Earth tides for
this study offer interesting results (Figs 12 and 13). The strongest correlation between
tremor and Earth tides occurs at La Push (Fig. 13a), most likely because of the tidal
loading present at that site which creates more pronounced Coulomb stresses. Of course,
La Push is not near the 40 km depth contour where most Cascadia tremor has been
located, but this can be ignored by comparing the correlation between tremor and ocean
tides at La Push and Sequim (which is near the 40 km depth contour)- both show
correlations near zero lag, and in fact, the correlation at Sequim has a peak at 0 days (Fig.
12b) while the correlation at La Push has a peak at –2 days (Fig. 12a). Therefore, it is
likely that if the Sequim tensor, for example, were taken with a large ocean tidal loading
effect, it would show a correlation with tremor similar to that of La Push.
The results shown in Figures 14 and 15 are similar to the cross-correlation results.
That is, the strongest evidence that tremor is related to Earth tides is found in the
comparison at La Push (Fig. 15a), with a probability of only 0.09 that this distribution of
tremor hours and days in relation to peak Coulomb stresses would be found with
uniformly distributed tremor. The more uniformly distributed results at Victoria, Sequim,
and Hoodsport could be due to the ocean tidal loading effect described above, or they
could be reflecting a reality that Earth tides do not affect tremor.
33
The correlations and methods presented here could use some improvements (see
below), but they still show a possible correlation between Earth tide stresses and tremor
occurrences. If this is the case, it would indicate that tremor can be thought of as slip on
a plane parallel to the Juan de Fuca plate under Washington state, and that this zone is
near a critical value of stress that would allow stress fluctuations due to Earth tides to
trigger a tremor episode.
Possibilities for Future Study
One problem with this study is that since the stress tensors used are calculated at
the surface of the Earth, it is difficult to extrapolate stresses to depths of 30 or 40 km
where tremor usually occurs. Ideally, stress tensors at these depths would be used, but
there are currently no established methods for doing this (Agnew, 2006; Wilcock, 2006).
It is possible that the effect of ocean tidal loading would extend inland from the ocean at
these depths, or, alternatively, the Earth tides alone would dominate. However, it is
likely that the stresses due to Earth tides would remain qualitatively the same (Creager,
2007), so the methods used in this study are still valid.
This study would also benefit from an extended time series of tremor. Since data
was only examined for the year 2006, only 77 days with tremor were recorded. Compare
this with other studies that examined correlations between Earth tides and earthquakes
which used data sets containing anywhere from 988 to 13,042 earthquakes (Tsuruoka et
al., 1995; Vidale et al., 1998; McNutt and Heaton, 1981; Wilcock, 2001). Larger data
sets help to reduce the effects of outliers, and provide more reliable results.
34
Since Coulomb stresses due to tides vary daily, comparisons between tremor and
Coulomb stresses should be performed on a daily scale. If the two were related, one
would expect to see more tremor occurring during hours of low tide (corresponding to
high Coulomb stresses) and less tremor occurring during hours of high tide
(corresponding to low Coulomb stresses). Although in this study I did record the hours
of tremor episodes, the resolution was not on a fine enough scale.
A better way to examine the relationship between tremor and Coulomb stresses on
a daily scale would be to examine the amplitude of tremor during the 14-month episodic
tremor and slip events, in which tremor is occurring almost continuously. Since the
tremor amplitude would be a continuous data set, it would be simple to compare it with
variations in the tidal cycle. A good data set to use would be from the most recent
episodic tremor and slip event that occurred in late January and early February of 2007.
CONCLUSIONS
In this study I determined that tremor in the Cascadia subduction zone occurred
during the year 2006 as a number of small events usually lasting fewer than three days.
Furthermore, after comparison with Coulomb stresses due to Earth tides and ocean tidal
loading, there is some enticing evidence in support of the hypothesis that tidal stresses
can trigger tremor. However, one would have to perform an analysis with larger data sets
and more rigorous statistical tests in order to confirm this hypothesis with certainty.
35
ACKNOWLEDGEMENTS
This project was started in the summer of 2006 as part of an IRIS (Incorporated
Research Institutions for Seismology) internship at the University of Washington. I
worked closely with my faculty advisor at the University of Washington, Ken Creager,
and a graduate student, Aaron Wech. My advisor at Carleton was Sarah Titus, who was
very helpful. Duncan Agnew (UCSD- Scripps Institution of Oceanography) and William
Wilcock (University of Washington) provided code and advice on Earth tides and ocean
tidal loading. I am very grateful to all of these people (and organization), as well as
anyone else who provided help and/or support along the way.
36
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39
Appendix A: Stress tensor calculations Plane defined by three vectors: v1 (strike) = <cos a, sin a, 0>; v2 (dip) = <-sin a cos b, cos a cos b, sin b>; v3 (normal) = <sin a sin b, -cos a sin b, cos b>; vslip (in direction of plate motion) = slip-dot(slip,v3)*v3, where slip=<cos(dir) sin(dir) 0>; a = angle of strike, and b = angle of dip, dir = plate motion direction T=dot(σ,v3) -Where T=traction vector, σ=stress tensor and dot() means dot product; |σn|=dot(T,v3) -(normal stress) [| | means magnitude]; |σs|=dot(T,vslip) -(shear stress); Check if: |σs| > μ(-|σn|-Pf) where μ is the coefficient of friction (usually from .5-1) and Pf is pore pressure;
40
Appendix B: Tremor events for 2006 Event Starting
date Number of days in event
Hours/day Event Starting date
Number of days in event
Hours/day
1 1/6/2006 2 1.5 19 6/6/2006 1 0.5 13.5 20 6/10/2006 2 0.25
2 1/14/2006 4 9.5 1.75 3 21 6/18/2006 2 3 0.5 1.75 1.5 22 6/23/2006 1 1.5
3 1/20/2006 3 2.5 23 7/11/2006 1 1 2.5 24 7/19/2006 3 1.25 1.5 0
4 1/27/2006 1 1 0.55 2/4/2006 1 0.5 25 7/26/2006 1 1.56 2/7/2006 2 1.75 26 8/3/2006 1 2.25
2.25 27 8/8/2006 3 4.57 2/12/2006 1 1 138 2/21/2006 1 0.5 4
2/25/2006 1 2.75 28 8/18/2006 2 1.59 3/3/2006 3 5 1
3 29 8/21/2006 2 0.5 2.5 2
10 3/16/2006 3 0.5 30 9/11/2006 5 7 1.5 11.5 1 16.5
11 3/27/2006 3 1 18 1.75 8 11 31 10/7/2006 1 0.5
12 4/1/2006 1 2.25 32 10/10/2006 5 4.513 4/9/2006 2 3 0
7.25 0.514 4/15/2006 4 2.25 0
4.25 1 12.5 33 11/4/2006 1 1 4 34 11/9/2006 4 0.5
15 4/23/2006 1 3.5 616 5/1/2006 2 2.5 3
2 117 5/13/2006 3 11.25 35 11/29/2006 1 0.75
4.5 36 12/5/2006 1 0.75 0.5 37 12/18/2006 1 1.25
18 5/26/2006 2 0.25 38 12/30/2006 2 1 0.25 1.75
41
Appendix C: Matlab code Figures2.m- Assemble all data and make figures %Assemble all data and make figures load tremor2006.txt; load tides2006a.txt; load tides2006b.txt; load tensor2006a.txt; %load tensor2006a2.txt; load tensor2006b.txt; load tensor2006c.txt; load tensor2006d.txt; tremor=tremor2006; tidesa=tides2006a; tidesb=tides2006b; tensa=tensor2006a; %tensa2=tensor2006a2; tensb=tensor2006b; tensc=tensor2006c; tensd=tensor2006d; [sheara,normala,failuresa,maxfailurea]=tremtenstide5(tensa,315,11); %[sheara2,normala2,failuresa2,maxfailurea2]=tremtenstide5(tensa2,315,11); [shearb,normalb,failuresb,maxfailureb]=tremtenstide5(tensb,345,11); [shearc,normalc,failuresc,maxfailurec]=tremtenstide5(tensc,315,11); [sheard,normald,failuresd,maxfailured]=tremtenstide5(tensd,0,11); tbig=[1:1/48:365.99]; intday=[1:365]; for i=1:365; i1=find(abs(intday(i)+.5-tbig)<14/24);%gets indices of tbig corresponding %to integer days with 2 hours of %space on either side to account %for tidal phase shift rangea(i)=max(tidesa(i1))-min(tidesa(i1)); rangeb(i)=max(tidesb(i1))-min(tidesb(i1)); i2=find(abs(intday(i)+.5-tbig)<.5); %gets indices of tbig corresponding %to integer days mintidea(i)=min(tidesa(i2)); mintideb(i)=min(tidesb(i2)); end %determine number of tremor hours/days within bins around maximum tides [daysA,tremhA,tremdA,testA]=rangetest5(maxfailurea); [daysB,tremhB,tremdB,testB]=rangetest5(maxfailureb); [daysC,tremhC,tremdC,testC]=rangetest5(maxfailurec);
42
[daysD,tremhD,tremdD,testD]=rangetest5(maxfailured); [daysAt,tremhAt,tremdAt,testAt]=rangetest5(rangea); [daysBt,tremhBt,tremdBt,testBt]=rangetest5(rangeb); %perform crosscorrelations of tremor and tidal series [Xat,LAGSat]=montecorr(tremor,rangea); [Xbt,LAGSbt]=montecorr(tremor,rangeb); [Xa,LAGSa]=montecorr(tremor,maxfailurea); [Xb,LAGSb]=montecorr(tremor,maxfailureb); [Xc,LAGSc]=montecorr(tremor,maxfailurec); [Xd,LAGSd]=montecorr(tremor,maxfailured); %plot cross-correlations figure(41);%tremor and ocean tidal range subplot(2,1,1); plot(LAGSat,Xat,'.-'); subplot(2,1,2); plot(LAGSbt,Xbt,'.-'); figure(42); subplot(2,2,1);%La Push plot(LAGSa,Xa,'.-'); subplot(2,2,2);%Victoria plot(LAGSc,Xc,'.-'); subplot(2,2,3);%Sequim plot(LAGSb,Xb,'.-'); subplot(2,2,4);%Hoodsport plot(LAGSd,Xd,'.-'); t1=14593;%Nov. 1 (day 305) t2=14641;%Nov. 2 (day 306) t3=16033;%Dec. 1 (day 335) %plot number of tremor hours/days in bins around tidal maxima clear abox; clear bbox; clear cbox; clear dbox; dhscale=3.37;%to get days and hours on the same scale scaled=.211;%77/365, likelihood of tremor/day %Tremor and ocean tides figure(31);clf; subplot(2,1,1); abox(1,:)=tremdAt*dhscale;abox(2,:)=tremhAt; bar(daysAt,abox','grouped');hold on; plot(daysAt,testAt*scaled*dhscale,'k:'); axis([-9 9 0 50]); ylabel('Tremor hours [r] (day*4 [b])'); title('La Push');
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subplot(2,1,2); bbox(1,:)=tremdBt*dhscale;bbox(2,:)=tremhBt; bar(daysBt,bbox','grouped');hold on; plot(daysBt,testBt*scaled*dhscale,'k:'); axis([-9 9 0 50]); ylabel('Tremor hours [r] (day*4 [b])'); title('Sequim'); xlabel('Days from peak tidal range'); %Tremor and Coulomb stresses clear abox; clear bbox; clear cbox; clear dbox; figure(32);clf; subplot(2,2,1);%La Push abox(1,:)=tremdA*dhscale;abox(2,:)=tremhA; bar(daysA,abox','grouped');hold on; plot(daysA,testA*scaled*dhscale,'k:'); axis([-9 9 0 50]); ylabel('Tremor hours [r] (day*4 [b])'); title('La Push'); subplot(2,2,2);%Victoria cbox(1,:)=tremdC*dhscale;cbox(2,:)=tremhC; bar(daysC,cbox','grouped');hold on; plot(daysC,testC*scaled*dhscale,'k:'); axis([-9 9 0 50]); title('Victoria'); subplot(2,2,3);%Sequim bbox(1,:)=tremdB*dhscale;bbox(2,:)=tremhB; bar(daysB,bbox','grouped');hold on; plot(daysB,testB*scaled*dhscale,'k:'); axis([-9 9 0 50]); ylabel('Tremor hours [r] (day*4 [b])'); title('Sequim'); xlabel('Days from peak Coulomb stress'); subplot(2,2,4); dbox(1,:)=tremdD*dhscale;dbox(2,:)=tremhD; bar(daysD,dbox','grouped');hold on; plot(daysD,testD*scaled*dhscale,'k:'); axis([-9 9 0 50]); title('Hoodsport'); xlabel('Days from peak Coulomb stress'); %Tides vs. stresses at La Push on Nov. 1(stresses in the ocean) figure(12);clf; subplot(2,1,1);plot(tbig(t1:t2),tidesa(t1:t2),'b'); axis([305 306 -1 1]); ylabel('Tide height (m)'); title('Tide height at La Push, Nov. 1'); subplot(2,1,2);plot(tbig(t1:t2),sheara(t1:t2),'r');hold on; plot(tbig(t1:t2),normala(t1:t2),'g--');
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plot(tbig(t1:t2),failuresa(t1:t2),'k-.'); axis([305 306 -10^4 10^4]); ylabel('Stress (Pa)'); title('Shear, normal, and max Coulomb at La Push'); xlabel('Day of 2006'); %Tides vs. stresses at Sequim on Nov. 1(stresses on land) figure(13);clf; subplot(2,1,1);plot(tbig(t1:t2),tidesb(t1:t2),'b'); %axis([305 306 -1 1]); ylabel('Tide height (m)'); title('Tide height at Sequim, Nov. 1'); subplot(2,1,2);plot(tbig(t1:t2),shearc(t1:t2),'r');hold on; plot(tbig(t1:t2),normalc(t1:t2),'g--'); plot(tbig(t1:t2),failuresc(t1:t2),'k-.'); %axis([305 306 -300 300]); ylabel('Stress (Pa)'); title('Shear, normal, and max Coulomb at Sequim'); xlabel('Day of 2006'); %Tides vs. stresses at La Push for November (stresses in the ocean) figure(13);clf; subplot(2,1,1);plot(tbig(t1:t3),tidesa(t1:t3),'b'); axis([305 335 -2 2]); ylabel('Tide height (m)'); title('Tide height at La Push for November'); subplot(2,1,2);plot(tbig(t1:t3),sheara(t1:t3),'r');hold on; plot(tbig(t1:t3),normala(t1:t3),'g:'); plot(tbig(t1:t3),failuresa(t1:t3),'k-.'); axis([305 335 -2*10^4 2*10^4]); ylabel('Stress (Pa)'); title('Shear, normal, and max Coulomb at La Push'); xlabel('Day of 2006'); %Tides vs. stresses at Sequim for November (stresses in the ocean) figure(14);clf; subplot(2,1,1);plot(tbig(t1:t3),tidesb(t1:t3),'b'); axis([305 335 -2 2]); ylabel('Tide height (m)'); title('Tide height at Sequim for November'); subplot(2,1,2);plot(tbig(t1:t3),shearb(t1:t3),'r');hold on; plot(tbig(t1:t3),normalb(t1:t3),'g:'); plot(tbig(t1:t3),failuresb(t1:t3),'k-.'); axis([305 335 -700 700]); ylabel('Stress (Pa)'); title('Shear, normal, and max Coulomb at Sequim'); xlabel('Day of 2006'); %Plot tremor figure(21);%tremor bar(tremor,'k'); axis([1 365 0 20]); title('Tremor during 2006'); xlabel('Day of 2006'); ylabel('Tremor hours');
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%Plot tremor with both tide series figure(22);clf%tremor with tidal range and Coulomb stress bar(intday,tremor,'k');hold on; plot(intday,rangeb*2+8,'b'); plot(intday,maxfailured/30,'r--'); axis([1 365 0 20]); title('Tremor, tidal range (sequim) and max Coulomb stress (hoodsport)'); xlabel('Day of 2006'); ylabel('Tremor hours'); tremtenstide5.m- Calculates Coulomb stresses at given point function [sheardir,normal,failures,maxfailure]=tremtenstide4(tensor,strike,dip); %Determine Coulomb stresses at given location a=strike; %slab strike b=dip; %slab dip mu=.6; %coefficient of friction (unitless?) Pf=0; %pore pressure (MPa) dir=55; %slab movement direction (from N) %Slab geometry vectors v1=[cosd(a), sind(a), 0]; %strike v2=[-sind(a)*cosd(b), cosd(a)*cosd(b), sind(b)];%dip v3=[sind(a)*sind(b), -cosd(a)*sind(b), cosd(b)];%normal slip=[cosd(dir) sind(dir) 0];%direction of plate motion at surface vslip=slip-dot(slip,v3)*v3; %projection onto plane vslip=vslip/sqrt(dot(vslip,vslip));%unit vectored %Compute failure amount for each point failures=[1:length(tensor)]; for i=1:length(tensor); row=i;%testing S= [tensor(row,1), tensor(row,6), tensor(row,5); tensor(row,6), tensor(row,2), tensor(row,4); tensor(row,5), tensor(row,4), tensor(row,3);]; t2 = S*v3' ; ns2=v3*t2; ss2=v2*t2; ss3=vslip*t2; shear(i)=ss2; sheardir(i)=ss3;%ss in plate motion direction normal(i)=ns2; x=ss3+mu*ns2; failures(i)=x; end; maxfailure=[1:365]; for iday=1:365; maxf=failures((iday-1)*48+1);
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for ihour=1:48; index=(iday-1)*48+ihour; if failures(index)>maxf; maxf=failures(index); en d; end; maxfailure(iday)=maxf; end rangetest5.m- put tremor hours/days in bins around tidal maxima function [days,tremh,tremd,test] = rangetest4(series); %takes a peak series and determines the amount of tremor in the days around %them %load tensor2006d.txt; %tensd=tensor2006d; %[sheard,normald,failuresd,maxfailured]=tremtenstide5(tensd,0,11); %series=maxfailured; load tremor2006.txt; tremor=tremor2006; tbig=[1:1/48:365.99]; intday=[1:365]; A=diff(series); ind=find(A(1:363)>0 & A(2:364)<0); ind(length(ind)+1)=ind(length(ind))+14; %for i=1:length(ind); % peaks(ind(i)+1)=1; %end peaks=ind+1; maxNum=max(diff(ind)); index=0; for j=1:maxNum; tremh(j)=0; tremd(j)=0; test(j)=0; end %go through year, put tremor hours into day boxes for i=1:365; if i>=peaks(index+1);%keep the window between two syzygys index=index+1; end if index==0; d1=-i; else d1=peaks(index)-i;
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end d2=peaks(index+1)-i; if abs(d1)<d2; d=d1; else d=d2; end mid=ceil(maxNum/2); tremh(mid+d)=tremh(mid+d)+tremor(i); if tremor(i)~=0; tremd(mid+d)=tremd(mid+d)+1; end test(mid+d)=test(mid+d)+1; end days=-floor(maxNum/2):floor(maxNum/2); if (length(days)>length(tremd)); days=days(2:length(days)); end montecorr.m- Returns a cross-correlation of two data sets using the matlab command xcorr (different version than xcor) function [X,LAGS] = montecorr(totaltrem,range); %takes a tremor daily series and tide range daily series and xcorr's the s* out of them intday=[1:365]; time_clip=5; g=range-mean(range); f=totaltrem-mean(totaltrem); %f = totaltrem; f=min(f,time_clip); f=f-mean(f); [X,LAGS]=xcorr(g,f,60,'coeff');
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